Error Correction With Fast Syndrome Calculation

ABSTRACT

Error correction is proposed, wherein, on the basis of a data word, a syndrome calculation is carried out with a matrix M on the basis of a matrix H of a code, and, if the result of the syndrome calculation reveals that the data word is erroneous, the result of the syndrome calculation is transformed by means of a linear mapping. Next, an error vector is determined on the basis of the result of the linear mapping by means of an efficient error correction algorithm and the erroneous data word is corrected on the basis of the error vector.

TECHNICAL FIELD

The approach described here relates to solutions for error processing,comprising, e.g., the detection and/or correction of errors.

SUMMARY

An object of the invention consists in avoiding disadvantages of knownsolutions for correcting errors, and in particular enabling an efficientcorrection of errors.

This object is achieved in accordance with the features of theIndependent claims. Preferred embodiments can be gathered from thedependent claims, in particular.

These examples proposed herein can be based on at least one of thefollowing solutions. In particular, combinations of the followingfeatures can be used to achieve a desired result. The features of themethod can be combined with (an) arbitrary feature(s) of the device orof the circuit, or vice versa.

In order to achieve the object, a method for error correction isspecified,

-   wherein, on the basis of a data word, a syndrome calculation is    carried out with a matrix M on the basis of a matrix H of a code,-   wherein, if the result of the syndrome calculation reveals that the    data word is erroneous, the result of the syndrome calculation is    transformed by means of a linear mapping,-   wherein an error vector is determined on the basis of the result of    the linear mapping by means of an efficient error correction    algorithm,-   wherein the erroneous data word is corrected on the basis of the    error vector.

The data word can be arbitrary information, for example a predefinednumber of bits or bytes.

In one development, no correction is effected if the fact that no errorwas detected was determined on the basis of the result of the syndromecalculation.

In one development, the matrix M used for the syndrome calculation hasthe following properties:

-   each row of the matrix M has the fewest possible ones,-   the matrix M overall has the fewest possible ones.

In one development, the matrix M used for the syndrome calculation isdetermined on the basis of the matrix H of the code as follows:

-   linear combinations are determined for the row vectors of the matrix    H,-   a Hamming weight is determined for each linear combination,-   row vectors of the matrix M are determined on the basis of the    linear combinations prioritized according to their rising Hamming    weight, coefficients of the linear combinations respectively    selected determining rows of a matrix J.

In one development,

-   non-trivial linear combinations are determined for the row vectors    of the matrix H,-   the Hamming weight is determined for each linear combination,-   the linear combinations are grouped according to their Hamming    weight,-   linear combinations are selected in tum with rising Hamming weight    and a row of the matrix M is determined on the basis of each    selected linear combination and a row of the matrix J is determined    on the basis of the coefficients of the selected linear combination,-   the linear combinations are selected with rising Hamming weight    until all the rows of the matrix M and all the rows of the matrix J    have been determined.

In one development, the linear mapping is based on the inverse matrixJ⁻¹.

In one development, the matrix J⁻¹ is determined such that it has thefewest possible ones per row.

Furthermore, a device for error correction is proposed, comprising aprocessing unit, configured for carrying out the following steps:

-   on the basis of a data word, carrying out a syndrome calculation    with a matrix M on the basis of a matrix H of a code,-   if the result of the syndrome calculation reveals that the data word    is erroneous, transforming the result of the syndrome calculation by    means of a linear mapping,-   determining an error vector on the basis of the result of the linear    mapping by means of an efficient error correction algorithm,-   correcting the erroneous data word on the basis of the error vector.

The processing unit mentioned here can be embodied in particular as aprocessor unit and/or an at least partly hardwired or logical circuitarrangement which is configured for example in such a way that themethod as described herein is able to be carried out. Said processingunit can be or comprise any kind of processor or computer withappropriately required peripherals (memory, input/output interfaces,input-output devices, etc.).

The above explanations concerning the method apply, mutatis mutandis, tothe device. The respective device can be implemented in one component orin a manner distributed among a plurality of components.

In one development, the device and/or the processing unit are/isconfigured such that no correction is effected if the fact that no errorwas detected was determined on the basis of the result of the syndromecalculation.

In one development, the matrix M used for the syndrome calculation hasthe following properties:

-   each row of the matrix M has the fewest possible ones,-   the matrix M overall has the fewest possible ones.

In one development, the matrix M used for the syndrome calculation isdetermined on the basis of the matrix H of the code as follows:

-   determining linear combinations for the row vectors of the matrix H,-   determining a Hamming weight for each linear combination,-   determining row vectors of the matrix M on the basis of the linear    combinations prioritized according to their rising Hamming weight,    coefficients of the linear combinations respectively selected    determining rows of a matrix J.

In one development, the device and/or the processing unit are/isfurthermore configured for

-   determining non-trivial linear combinations for the row vectors of    the matrix H,-   determining the Hamming weight for each linear combination,-   grouping the linear combinations according to their Hamming weight,-   selecting linear combinations in turn with rising Hamming weight, a    row of the matrix M being determined on the basis of each selected    linear combination and a row of the matrix J being determined on the    basis of the coefficients of the selected linear combination,-   selecting the linear combinations with rising Hamming weight until    all the rows of the matrix M and all the rows of the matrix J have    been determined.

In one development, the device and/or the processing unit are/isconfigured such that the linear mapping is based on the Inverse matrixJ⁻¹.

In one development, the device and/or the processing unit are/isconfigured such that the matrix J⁻¹ is determined such that it has thefewest possible ones per row.

Additionally, solutions described herein can take account of or comprisethe following approaches: a method for error correction,

-   wherein a syndrome calculation is carried out in a code domain of a    second code,-   wherein an efficient error correction algorithm is carried out in a    code domain of a first code.

The code domain is determined for example by a vector space of the code.The code is preferably an error detecting and/or error correcting code.The first code has an efficient error correction algorithm. Preferably,the first code is a code for which such an efficient error correctionalgorithm is known. By way of example, the first code is one of thefollowing codes: a Hamming code, a BCH code, a Reed-Muller code, aSimplex code, a Golay code or a Goppa code.

Changing between the code domains makes it possible to carry out thesyndrome calculation more efficiently in the code domain of the secondcode and nevertheless to be able to use the efficient error correctionalgorithm of the first code (after changing back to the code domainthereof).

In one development, between the syndrome calculation and carrying outthe efficient error correction algorithm, a transition between the codedomains is carried out by means of at least one linear mapping.

In one development,

-   a data word is received in the code domain of the first code and is    converted into the code domain of the second code by means of a    first linear mapping,-   the syndrome calculation is carried out in the code domain of the    second code on the basis of the result of the first linear mapping    and, if the result of the syndrome calculation reveals that the data    word is erroneous, the result of the syndrome calculation is    converted into the code domain of the first code by means of a    second linear mapping,-   an error vector is determined on the basis of the result of the    second linear mapping in the code domain of the first code by means    of the efficient error correction algorithm,-   the erroneous data word is corrected on the basis of the error    vector.

In one development,

-   the syndrome calculation is carried out in the code domain of the    second code on the basis of a received data word and, if the result    of the syndrome calculation reveals that the data word is erroneous,    the result of the syndrome calculation is converted into the code    domain of the first code by means of a second linear mapping.-   an error vector of the code domain of the first code is determined    on the basis of the result of the second linear mapping in the code    domain of the first code by means of the efficient error correction    algorithm,-   the error vector of the code domain of the first code is converted    into an error vector of the code domain of the second code by means    of a first linear mapping,-   the erroneous data word is corrected on the basis of the error    vector of the code domain of the second code.

In one development,

-   the first code is determined by a matrix H,-   the second code is determined by a matrix K,-   the first linear mapping P and the second linear mapping L are    determined such that the following holds true:-   H = L ⋅ K ⋅ P.

In one development, the matrix K is a check matrix comprising a unitmatrix.

In one development, the first linear mapping comprises a permutation.

In one development, no correction is effected if the fact that no errorwas detected was determined on the basis of the result of the syndromecalculation.

An exemplary device for error correction can comprise a processing unitconfigured for carrying out the method described herein.

Moreover, a computer program product is proposed which is directlyloadable into a memory of a digital computer, comprising program codeparts suitable for carrying out steps of this method.

Furthermore, the problem mentioned above is solved by means of acomputer-readable storage medium, e.g., of an arbitrary memory,comprising instructions (e.g., in the form of program code) which areexecutable by a computer and which are suitable for the purpose of thecomputer carrying out steps of the method described here.

The above-described properties, features and advantages of thisinvention and the way in which they are achieved are described below inassociation with a schematic description of exemplary embodiments whichare explained in greater detail in association with the drawings. Inthis case, identical or identically acting elements may be provided withidentical reference signs for the sake of clarity.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows an exemplary flow diagram for elucidating a mode ofoperation of a decoder for determining an error vector e.

FIG. 2 shows an exemplary circuit diagram for a redistribution.

FIG. 3 shows a code domain of a first code and a code domain of a secondcode, a transition between the code domains being effected by means of(e.g., linear) transformation and an efficient error correction beingachieved by the syndrome calculation being effected in the code domainof the second code and the efficient error correction algorithm beingeffected in the code domain of the first code.

FIG. 4 shows an alternative example to FIG. 3 for the efficient errorcorrection.

FIG. 5 shows an exemplary diagram for elucidating the efficient syndromecalculation in particular by means of hardware.

DETAILED DESCRIPTION

Each binary r×nmatrix H of rank r defines a binary linear code C havingthe length n and the dimension k=n-r. This code is the null space of thematrix H, i.e.

$C = \left\{ {\overset{\rightarrow}{c} \in \text{GF}(2)^{n}:H{\overset{\rightarrow}{c}}^{T} = 0} \right\}$

with

-   a code word c (also designated as vector c) of the code C,-   the finite field GF(2) = {0,1} of order two,-   GF(2)^(n) as n-dimensional vector space (Galois field) of all binary    row vectors having the length n, and-   c ^(T) as the transpose of the vector (code word) c.

The matrix H is called the check matrix of the code C. The code C isuniquely defined by the check matrix H. The code is an error detectingand/or error correcting code, for example.

A weight (Hamming weight) w(y) of a binary vector y is defined as thenumber of ones in the binary vector y. Each linear code contains thenull vector 0, having the weight null: w(0)= 0.

The smallest weight of all vectors c of a linear code C that aredifferent than the null vector is designated as the minimum distance ofthe code C:

$d = \min\left( {w\left( \overset{\rightarrow}{c} \right):\overset{\rightarrow}{c} \in C\backslash\left\{ \overset{\rightarrow}{0} \right\}} \right)$

A linear code having the length n, the dimension k and the minimumdistance d is designated as (n, k, d) code.

With a code having the minimum distance d, in principle all t-bit errorsare correctable, where the following holds true: 0 ≤ t < d / 2 .

In this case, correctable means, in particular, that the t errorpositions in the received erroneous code word are uniquely determined.These error positions are not calculable here in all cases (inparticular within a predefined time duration).

The error positions are determinable within a predefined time durationonly if an efficient error correction algorithm is known for the presentcode. This is the case for very few linear codes. Codes which have anefficient error correction algorithm are, e.g., the following codes:Hamming code, BCH code, Reed-Muller code, simplex code, Golay code,Goppa code.

In practice, e.g., on computer chips, use is made of codes having anefficient error correction algorithm.

Example 1: Consider a linear code having the length n = 100 with aminimum distance d = 21. A 10-bit error occurs during data transmission.No efficient error correction algorithm is known for the code. In orderto determine the error, all

100!/(90! * 10!) = 17310309456440

possible error patterns have to be tried out. The syndrome is equal tozero for exactly one error pattern: this error pattern corresponds tothe 10-bit error that has occurred. For all other error patterns, thesyndrome is not equal to zero. Therefore, approximately 17 billionsyndrome calculations are required to determine the 10-bit error.

Example 2: The simplex code having a length n = 127 has a minimumdistance d = 64. There is an efficient error correction algorithm forthis code. A 30-bit error occurs during data transmission. The errorcorrection algorithm (implemented in hardware as an electronic circuitcomprising 120 flip-flops and XOR and majority gates) allows the 30-biterror to be determined in 127 clock cycles of a processor unit (CPU).The error correction requires only a single syndrome calculation for theinitialization of the circuit.

Syndrome

Most error correction algorithms require a syndrome vector (alsoreferred to as syndrome) as input. The syndrome vector is calculatedfrom the received data word with the aid of the check matrix H.

If c is the transmitted code word and y is the associated received dataword, then the vector y is a (possibly erroneous) version of thetransmitted code word c. The syndrome of y is defined by

$\begin{matrix}{S\left( \overset{\rightarrow}{y} \right) = H \cdot {\overset{\rightarrow}{y}}^{T}.} & \text{­­­(1)}\end{matrix}$

If no error has occurred during transmission (or during storage), i.e.,if y = c, then the syndrome is equal to zero, i.e., the syndrome vectorS(y) is identical to the null vector 0. By contrast, if a (detectable)error has occurred, then it holds true that: S(y) ≠ 0.

FIG. 1 shows an exemplary flow diagram for elucidating a mode ofoperation of a decoder for determining an error vector e.

A code word c is determined in a step 101 and is transmitted via achannel 102 and is received as a data word (also referred to as datavector) y by the decoder in a step 103. In a step 104, the syndromecalculation is effected in accordance with equation (1).

If the syndrome is equal to zero, then the procedure branches to a step105; the received data word is identical to the transmitted code word (y= c). In this case, it is assumed that no transmission error hasoccurred; the received data word y is accepted as the transmitted(correct) code word c.

If the syndrome is not equal to zero, then the procedure branches fromstep 104 to a step 106: since the received data word is different thanthe transmitted code word (y ≠ c), an error correction algorithm is usedto determine the error vector e, such that the following holds true:

$\begin{matrix}{\overset{\rightarrow}{y} + \overset{\rightarrow}{e} = \overset{\rightarrow}{c}.} & \text{­­­(2)}\end{matrix}$

The error correction algorithm calculates error positions from thesyndrome vector S(y) and outputs them in the form of the error vector e.The received data word y is then corrected by a vector

$\overset{\rightarrow}{y} \oplus \overset{\rightarrow}{e}$

being calculated, where “⊕” corresponds to an exclusive-OR operation.This results in a corrected code word

$\begin{matrix}{{\overset{\rightarrow}{c}}_{corr} = \overset{\rightarrow}{y} \oplus \overset{\rightarrow}{e},} & \text{­­­(3)}\end{matrix}$

which is assumed to be identical to the transmitted code word c.

The syndrome calculation is preferably always performed, whereas theerror correction is only required if the syndrome is not equal to zero.

Example 3: A two-error-correcting code having the length n = 40 with aminimum distance d = 5 is assumed by way of example. 100000 code wordsare transmitted. A bit error probability is p = 0.001. Consequently, onaverage approximately 96000 code words are transmitted without errors.In the case of approximately 4000 code words, a 1-bit error or a 2-biterror occurs during transmission, these being corrected automatically.Just a single transmission involves the occurrence of a multi-bit error(i.e., three or more bits are erroneous) in the transmitted code word.Consequently, on average one of the 100 000 received data words cannotbe corrected. In 96% of the cases the syndrome is equal to zero and noerror correction is necessary. In 4% of the cases the syndrome is notequal to zero and it is only then that the error correction algorithm Isrequired.

Since the syndrome calculation is always carried out, but the errorcorrection is only carried out sometimes, it is expedient to implementthe syndrome calculation efficiently. In this way the computationduration is shortened and the power consumption of the decoder isreduced.

Efficient Syndrome Calculation in Software Canonical Check Matrix

Check matrices of the form

$\begin{matrix}{K = \left( {I,A} \right),} & \text{­­­(4)}\end{matrix}$

enable an efficient syndrome calculation, for example by means ofsoftware. In this case, I represents the unit matrix.

The columns of the unit matrix I are unit vectors, i.e., vectors havingonly a single one and otherwise all zeroes. The canonical matrix is thussparsely populated, i.e., it contains relatively few ones overall.

If the check matrix K is used for the syndrome calculation, i.e., acolumn vector

$\begin{matrix}{Z\left( \overset{\rightarrow}{y} \right) = K \cdot {\overset{\rightarrow}{y}}^{T}} & \text{­­­(5)}\end{matrix}$

is calculated and used as syndrome, then for the calculation of thecoordinates z_(i) of the syndrome

$\begin{matrix}{Z\left( \overset{\rightarrow}{y} \right) = \left( {z_{1},\ldots,z_{n - k}} \right)^{T}} & \text{­­­(6)}\end{matrix}$

only few coordinates y_(j) of the vector y = (y₁,..., y_(n)) have to beadded together (in the Galois field GF(2) such an addition correspondsto an exclusive-OR operation, also referred to as XOR operation) becausethe canonical matrix K contains few ones per row.

Consequently, in the case of a hardware implementation, the logicaldepth of a circuit for calculating the coordinates of the syndrome withthe use of the sparsely populated matrix K is smaller than in the caseof the syndrome calculation using a more densely populated matrix.

The syndrome calculation by means of such a canonical matrix K isadvantageous in the case of a software implementation of the code aswell: the (n-k)×(n-k) unit matrix I at the foremost position in thecheck matrix K= (I,A) has the effect that in the syndrome calculationinitially only the last k coordinates y_(j) of the vector y influencethe individual row sums. The components of the resulting column vectorare determined solely from the last k coordinates of the vector y.Afterward, the column vector

(y₁, y₂, …, y_(n − k))^(T)

is added to this column vector in order to obtain the syndrome Z(y) inaccordance with equation (6).

For i = 1,..., n-k the output but z_(i) is at the same place as thecorresponding input bit y_(i). This is advantageous for the programmingsince, for example, the programming languages C or C++ process data wordby word: the bitwise addition of the two column vectors can thus beeffected by means of a single instruction.

Example 4: Consider the following canonical check matrix by way ofexample:

$\begin{matrix}{K = \left( {\begin{matrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{matrix}\quad\begin{matrix}1 & 0 & 1 \\1 & 1 & 1 \\1 & 1 & 0 \\0 & 1 & 1\end{matrix}} \right).} & \text{­­­(7)}\end{matrix}$

K is the canonical check matrix of a 1-bit-error-correcting linear codehaving the length n = 7 and the dimension k = 3. The syndrome of

$\overset{\rightarrow}{y} = \left( {y_{1},\ldots,y_{7}} \right)$

results as follows:

$Z\left( \overset{\rightarrow}{y} \right) = K \cdot {\overset{\rightarrow}{y}}^{T} = \begin{pmatrix}z_{1} \\z_{2} \\z_{3} \\z_{4}\end{pmatrix} = \begin{pmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{pmatrix} \oplus \begin{pmatrix}{y_{5} + y_{7}} \\{y_{5} + y_{6} + y_{7}} \\{y_{5} + y_{6}} \\{y_{6} + y_{7}}\end{pmatrix}.$

(end of example 4)

The check matrix H of a linear (n,k,d) code C is not uniquelydetermined. If J is an invertible (n-k)×(n-k) matrix, then

$\begin{matrix}{H^{*} = J \cdot H} & \text{­­­(8)}\end{matrix}$

is likewise a check matrix of the code: the matrices H and H∗ have thesame null space and the code is by definition identical to this vectorspace. A linear code C therefore has a plurality of different checkmatrices.

If the linear code Chas an efficient error correction algorithm whichcalculates the error vector e from a syndrome S(y) = H·y ^(T) inaccordance with equation (1), then the set of different check matricescontains a specific check matrix: this specific check matrix hasproperties which the error correction algorithm utilizes, and which theother check matrices do not have. Therefore, the error correctionalgorithm functions only in conjunction with this specific check matrix.

It is only for codes from the simplex family and for the first-orderReed-Muller code that the specific check matrix is already canonical,i.e., in the form (I,A).

In order to improve the efficiency of the syndrome calculation, it isdesirable to use a canonical matrix for the syndrome calculation.However, the efficient error correction algorithm demands that thesyndrome be calculated with the associated specific check matrix.

Generation of the Canonical Matrix

Let Hbe an r×nmatrix of rank r. A canonical r×n matrix can be determinedfrom this matrix H by carrying out elementary row transformations forthe rows of the matrix H. Elementary row transformations correspond to apermutation of rows of the matrix H and a bitwise addition in the Galoisfield GF(2) of one matrix row to another. Furthermore, columns of thematrix H can be permuted among one another.

By means of a finite number of such operations, a matrix Kof the type(I,A) is determined from the matrix H. In principle, any matrix H havinga full rank can be transformed into a canonical matrix.

Carrying out elementary row transformations is equivalent to multiplyingthe matrix H from the left by an invertible r×r matrix A. Thepermutation of columns of the matrix H is equivalent to multiplying thematrix H from the right by an n×n permutation matrix B.

A permutation matrix is a matrix which has exactly a single one andotherwise only zeroes in each of its rows and in each of its columns.

There are, then, an invertible r×r matrix A and an n×n permutationmatrix B, such that

$\begin{matrix}{K = A \cdot H \cdot B} & \text{­­­(9)}\end{matrix}$

holds true, the resulting matrix Khaving the canonical form.Consequently, the canonical matrix K is determined from the matrix H onthe basis of the two matrices A and B.

The matrices A and B are both invertible. The inverse matrices aredesignated hereinafter by L and P, respectively, in accordance with therelationships:

$\begin{array}{l}{L: =} \\{A^{- 1}\mspace{6mu} P} \\{: = \mspace{6mu} B^{- 1}}\end{array}$

Since B is a permutation matrix, the inverse matrix Pis also apermutation matrix.

Equation (9) is thus equivalent to

$\begin{matrix}{H = L \cdot K \cdot P.} & \text{­­­(10)}\end{matrix}$

The vector y = ( y₁,.... y_(n)) is a binary row vector of length n. Ifequation (10) is multiplied from the right by the column vector y ^(T),it follows that

$\begin{matrix}{H \cdot {\overset{\rightarrow}{y}}^{T} = L \cdot K \cdot P \cdot {\overset{\rightarrow}{y}}^{T}.} & \text{­­­(11)}\end{matrix}$

If it is then the case that the dimension k-n-r, the matrix H havingr=n-k rows and n columns has the rank r = n - k. This means that thenull space of the matrix H is a linear (n, k, d) code C and the matrix His a check matrix for the code C.

The left-hand side of equation (11) corresponds, in accordance withequation (1), to the syndrome S(y) of the vector y in the code Crelative to the check matrix H.

The right-hand side of equation (11) contains the vector Py ^(T). Sincethe matrix P Is an n×n permutation matrix, the vector Py ^(T) containsthe same coordinates as the vector y, just in a different order.

In the case of a realization in hardware, a mapping

$\begin{matrix}\left. {\overset{\rightarrow}{y}}^{T}\mapsto P \cdot {\overset{\rightarrow}{y}}^{T} = {{\overline{y}}^{\prime}}^{T} \right. & \text{­­­(12)}\end{matrix}$

in the form of a redistribution can advantageously be implemented(largely) cost-neutrally.

Example 5: Consider a permutation matrix

$\begin{matrix}{P = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{pmatrix}} & \text{­­­(13)}\end{matrix}$

and a vector

$\overset{\rightarrow}{y} = \left( {y_{1},y_{2},y_{3},y_{4}} \right).$

What follows therefrom is:

${{\overset{\rightarrow}{y}}^{\prime}}^{T} = P \cdot {\overset{\rightarrow}{y}}^{T} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{pmatrix} \cdot \begin{pmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{pmatrix} = \begin{pmatrix}y_{1} \\y_{3} \\y_{2} \\y_{4}\end{pmatrix} = \begin{pmatrix}{y^{\prime}}_{1} \\{y^{\prime}}_{2} \\{y^{\prime}}_{3} \\{y^{\prime}}_{4}\end{pmatrix}.$

FIG. 2 shows an exemplary circuit diagram that elucidates aredistribution from y to y′ in accordance with this example. (End ofexample 5)

Equation (11) can be formulated with equation (12) as follows:

$\begin{matrix}{H \cdot {\overset{\rightarrow}{y}}^{T} = L \cdot K \cdot P \cdot {\overset{\rightarrow}{y}}^{T} = l \cdot K \cdot {{\overset{\rightarrow}{y}}^{\prime}}^{T}.} & \text{­­­(14)}\end{matrix}$

The left-hand side of equation (14) contains the syndrome S(y)=Hy ^(T)in accordance with equation (1), which is required by the errorcorrection algorithm. The right-hand side of equation (14) contains thesyndrome

$Z\left( \left| {\overset{\rightarrow}{y}}^{\prime} \right) \right) = K \cdot {{\overset{\rightarrow}{y}}^{\prime}}^{T}.$

It is thus evident that the two syndromes S:=S(y) and Z:=Z(y′) arelinear transformations with respect to one another:

$\begin{matrix}{S = L \cdot Z.} & \text{­­­(15)}\end{matrix}$

Owing to the permuted matrix columns in the course of deriving thematrix Kfrom the matrix H, the matrix Kis not a check matrix for theoriginal code C. Instead the matrix K defines a new code C.

The old code C and the new code C have the same parameters n, k and dand the same weight distribution. However, they determine differentvector spaces (also referred to as code domains): a code word of thecode C is not necessarily also a code word of the code C. Thecalculations of the syndromes Sand Z thus relate to different codes.

Solution Example

FIG. 3 shows a code domain 301 of a first code and a code domain 302 ofa second code. In this context, code domains (vector space) means thatthe calculations within the domain are specific to the respective code.A transition between the code domains is effected by means of (e.g.,linear) transformation.

1. For the first code C there is an efficient error correction algorithmwhich is based on properties of a specific check matrix H and requiresthe syndrome H·y ^(T) in accordance with equation (1) for the correctionof the received data word y.

A message m to be transmitted is coded with the first code C in a mannercompatible with the matrix H, i.e., a row vector m is multiplied by agenerator matrix Gbelonging to the matrix H in order to calculate a codeword c:

$\overset{\rightarrow}{c} = \overset{\rightarrow}{m} \cdot G.$

This code word c is transmitted and received as a bit sequence y (alsoreferred to as received data word, data vector, input or input vector)at the other end of the channel. This bit sequence y constitutes theinput in the code domain 301 in FIG. 3 .

2. Derivation of the canonical matrix:

-   For the matrix H, the associated canonical matrix-   K = (I, A),-   the invertible square matrix L and the permutation matrix P are    calculated, such that the following holds true in accordance with    equation (10):-   H = L ⋅ K ⋅ P.

3. Redistribution:

-   For the received data word y, a row vector-   ${\overset{\rightarrow}{y}}^{\prime} = \overset{\rightarrow}{y} \cdot P^{T}$-   is calculated. This corresponds to a transition 303 from the first    code domain 301 to the second code domain 302 by means of the matrix    P.

4. Fast syndrome calculation 304:

-   The syndrome Z(y′) of y′ is calculated using the canonical matrix K:-   $Z = Z\left( {\overset{\rightarrow}{y}}^{\prime} \right) = K \cdot {{\overset{\rightarrow}{y}}^{\prime}}^{T}.$-   If Z = 0, i.e., the K syndrome is zero, then y is error-free. The    method can be ended.-   If Z≠0, i.e., the K syndrome is not equal to zero, then y is not    error-free; the procedure branches to the subsequent step (5.) and a    linear transformation 305 is then carried out.

5. Linear transformation 305:

-   The column vector Z is multiplied by the square matrix L. The    resulting column vector-   S = L ⋅ Z-   is identical to the required syndrome vector S = H · y ^(T). The    multiplication by L corresponds to a transition from the second code    domain 302 to the first code domain 301.

6. Error vector calculation 306:

By means of the error correction algorithm, on the basis of the syndromevector S, an error vector e is calculated and output.

7. Error correction:

On the basis of the error vector e, the correction can be carried out:the corrected code word c _(corr) results as

${\overset{\rightarrow}{c}}_{corr} = \overset{\rightarrow}{y} + \overset{\rightarrow}{e}.$

Since most arriving data words y are error-free, the program alreadyends with step (4.) in the majority of cases. It is relatively rare foran error to occur in which the K syndrome is not equal to zero, and forsteps 5. to 7. be carried out.

In step 4. the syndrome calculation is effected. The time required forthis has a crucial effect on the overall performance of the solutionpresented here.

It should additionally be noted that the syndrome calculation 304 iseffected in the code domain 302 of the second code, i.e., after thetransition 303 (the transformation by means of P), using the matrix K.This replaces a syndrome calculation without transformation (i.e.,within the code domain 301 of the first code) by means of the matrixH(not Illustrated in FIG. 3 ). The syndrome calculation 304is faster bycomparison therewith, and so the resulting additional complexity of thetransitions 303and 305is of secondary importance relative to the overallperformance of the solution presented.

Alternative Solution

An alternative solution is described below. The same prerequisitesinitially hold true: for a first linear (n,k,d) code there is anefficient error correction algorithm which is associated with a specificcheck matrix H.

For correction purposes, the error correction algorithm requires as aninput the syndrome S(y)= H·y ^(T) defined by the matrix H in accordancewith equation (1).

From the matrix H, an equivalent matrix K of the form K-(I,A)can bedetermined. In this case, “equivalent” means that invertible squarematrices L and Pof appropriate size exist such that the following holdstrue in accordance with equation (10):

H = L ⋅ K ⋅ P.

In this case, the matrix P is the permutation matrix.

If the matrix P Is not itself the unit matrix, then the matrices Hand Kdefine different codes. These codes are equivalent to one another sincethe code words of one code emerge from the code words of the other codeby rearrangement according to a fixed specification.

Considered as vector spaces, however, they are two different codes. Thefirst code is the null space of the matrix H and the second code is thenull space of the matrix K.

In this alternative solution, the message coding is effected by means ofthe second code, i.e., the code associated with the canonical checkmatrix K. That has the following advantages, in particular:

1. The coding of the message having a length of k-bits, in the presentcase as a vector

$\overset{\rightarrow}{m} = \left( {m_{1},\ldots,m_{k}} \right)$

can be carried out with the aid of the generator matrix G belonging tothe canonical matrix K. This coding is also designated by K coding.

The generator matrix G has the form G=(I,B) with the k×k unit matrix l.The matrix K has the form K-(I,A) with the (n-k)×(n-k) unit matrix l.Consequently, the generator matrix G is likewise a canonical matrix.

The following holds true for the matrices Gand K:

G ⋅ K^(T) = 0,

where 0 here denotes the k×(n-k) zero matrix (all matrix entries arezero).

The coding of the message m (i.e., the conversion of the message m intoa code word c) is achieved by multiplication by the generator matrix G,i.e.

$\overset{\rightarrow}{c} = \overset{\rightarrow}{m} \cdot G.$

Since the generator matrix G is canonical, the code word c calculatedwith it has the form

$\overset{\rightarrow}{c} = \left( {\overset{\rightarrow}{m},\overset{\rightarrow}{r}} \right).$

The code word c consists of the message vector m with the n-k redundancybits r appended thereto.

Consequently, in the second code (i.e., the null space of the canonicalmatrix K) both the syndrome calculation and the coding can be performedmore efficiently than in the first code (i.e., the null space of theerror correction matrix H).

2. FIG. 3 explained above shows an approach for coding the message m inthe first code, i.e., in the code with the efficient error correctionalgorithm. In this case, each received data word y is firstly permutedcoordinate by coordinate, i.e., the permutation in accordance with thematrix P is applied to each received data word.

The alternative solution described in the present case departs fromthis: the received data word y is already present in coded form in thecode with the canonical check matrix K with which the efficient syndromecalculation is also carried out. It can thus be fed unchanged (i.e.,without permutation) to the syndrome calculation. If the syndrome isequal to zero, no correction takes place (the received data word isregarded as correct and the method can be ended).

By contrast, if the K syndrome Z is not equal to zero, then it islinearly transformed using the square matrix L The transformed syndrome

S = L ⋅ Z

is fed in the error correction algorithm of the first code, whichcalculates an error vector e therefrom (by means of the first code). Inorder to determine the associated error vector e from the error vectore′ this associated error vector being required in the second code forthe error correction taking place there, the permutation in accordancewith the matrix P is applied to the error vector e′.

FIG. 4 illustrates this relationship on the basis of the code domain 301of the first code and the code domain 302 of the second code.

In the code domain 302, a K coding 401 of the message vector m into thecode word c is effected, which code word is transmitted via a channel402 and received as a data word y. For the received data word y, a fastsyndrome calculation 403 is effected in a manner comparable with thesyndrome calculation 304 from FIG. 3 :

The syndrome Z(y) of y is calculated using the canonical matrix K:

$Z = Z\left( \overset{\rightarrow}{y} \right) = K \cdot {\overset{\rightarrow}{y}}^{T}.$

If Z=0, i.e., the K syndrome is zero, then y is error-free. The methodcan be ended.

If Z≠0, i.e., the K syndrome is not equal to zero, then y is noterror-free. In this case, multiplication by the matrix L is used toeffect a linear transformation 404

S = L ⋅ Z,

which brings about a transition from the code domain 302 to the codedomain 301.

By means of an error correction algorithm 405 of the code domain 301,the error vector e is calculated on the basis of the syndrome vector Sand is transformed into the error vector e of the second code domain 302by means of the matrix P 406. The received data word y can be correctedon the basis of the error vector e:

${\overset{\rightarrow}{c}}_{corr} = \overset{\rightarrow}{y} + \overset{\rightarrow}{e}.$

As a result, the permutation of the solution shown in FIG. 4 isperformed less frequently than in the example shown in FIG. 3 . This isadvantageous particularly in the case of software implementations.

Efficient Syndrome Calculation in Particular by Means of Hardware

A matrix M which can for example be implemented in hardware and is usedfor the syndrome calculation is intended to have the followingproperties:

-   (A) Each row of the matrix M is intended to contain the fewest    possible ones.-   (B) The matrix M is intended overall to contain few ones.

Property (A) is important because the logical depth for the calculationof the individual syndrome coordinates is thus as small as possible. Thesmaller the logical depth of a circuit, the faster this circuit can beclocked.

Property (B) is important because fewer ones in the matrix means thatthe matrix can be implemented with a smaller number of XOR gates, whichnecessitates a smaller semiconductor area.

Consider a matrix H having r×n elements, which is a specific checkmatrix with respect to an error correction algorithm used. The errorcorrection algorithm requires as input the syndrome S(y)=H·y ^(T)calculated using the matrix H, where y is the vector of the receiveddata word.

With an arbitrary invertible r × r matrix J, a matrix

$\begin{matrix}{M = J \cdot H} & \text{­­­(16)}\end{matrix}$

is also a check matrix for the code.

If all possble invertible binary r×r matrices J are inserted in equation(16) (which leads to a very large number even for low values of r), thenthe matrix M runs through all existing check matrices of the code (whichis defined by the matrix H). These check matrices include matriceshaving the properties (A) and (B).

If equation (16) is multiplied by an inverse matrix J⁻¹ on both sides,it follows that:

$\begin{matrix}{J^{- 1} \cdot M = H.} & \text{­­­(17)}\end{matrix}$

If equation (17) is multiplied by y ^(T) from the right, this results inthe following:

$\begin{matrix}{J^{- 1} \cdot M \cdot {\overset{\rightarrow}{y}}^{T} = H \cdot {\overset{\rightarrow}{y}}^{T}} & \text{­­­(18)}\end{matrix}$

The right-hand side of equation (18) contains the H syndrome

$S\left( \overset{\rightarrow}{y} \right) = H \cdot {\overset{\rightarrow}{y}}^{T}.$

The left-hand side of equation (18) contains the syndrome M·y ^(T),calculated using the matrix M, multiplied by the matrix J⁻¹.

The inverse matrix J⁻¹ can be used by way of example in a hardwareimplementation which does not require the matrix J.

By way of example, the matrix J⁻¹ is designated by F. i.e., F =J⁻¹.

It is advantageous that the matrix J⁻¹ contains the smallest possiblenumber of ones per row. Therefore, a third property for an efficientsyndrome calculation in hardware reads as follows:

(C) The inverse matrix J⁻¹ with respect to the matrix J in accordancewith equation (16) is intended to contain the fewest possible ones perrow.

FIG. 5 shows an exemplary diagram for elucidating the efficient syndromecalculation by means of hardware. In this case, only one code domain 501(i.e., one vector space of the code C, for which there is also theefficient error correction algorithm) is assumed by way of example; atransition between different code domains is not required.

On the basis of the received data word y, a syndrome calculation 502 iseffected by means of the matrix M:

$W = W\left( \overset{\rightarrow}{y} \right) = M \cdot {\overset{\rightarrow}{y}}^{T}.$

If W=0, i.e., the M syndrome is zero, then y is error-free. The methodcan be ended.

If W≠0, i.e., the M syndrome is not equal to zero, then y is noterror-free, and a linear transformation 503 is carried out by means ofthe matrix F:

S = F ⋅ W.

On the basis of an error correction algorithm 504, an error vector e iscalculated on the basis of the syndrome vector S and the received dataword y can be corrected by means of the error vector e:

${\overset{\rightarrow}{c}}_{corr} = \overset{\rightarrow}{y} + \overset{\rightarrow}{e}.$

Calculation of Hardware-Optimized Matrices M and J-1

An example of the hardware-optimized determination of the matrices M andJ⁻¹ is given below.

Let h ₁,..., hr be the rows of the r × n matrix H.

1. (In particular) all 2^(r)-1 non-trivial linear combinations

$k_{1} \cdot {\overset{\rightarrow}{h}}_{1} + \ldots + k_{r} \cdot {\overset{\rightarrow}{h}}_{r}$

of the r row vectors are determined.

2. The Hamming weight of each linear combination is calculated and thelinear combinations are sorted according to ascending Hamming weight: atthe beginning of the list there are the linear combinations having theHamming weight 1 (if there are any), followed then by the linearcombinations having the Hamming weight 2, etc.

Linear combinations having the same Hamming weight can be combined ingroups. The arrangement of the linear combinations within a group can bechosen freely or according to a predefined scheme.

3. A linear combination

${\overset{\rightarrow}{v}}_{1} = a_{1} \cdot {\overset{\rightarrow}{h}}_{1} + \ldots + a_{r} \cdot {\overset{\rightarrow}{h}}_{r}$

is selected from the first group. The row vector v ₁ is thus the firstrow of the hardware-optimized check matrix M to be determined and(a₁,..., a_(r)) is the first row of the matrix J.

4. If present, a linear combination v ₂ ≠ v ₁ is selected from the firstgroup. If not present, the linear combination v ₂ is selected from thesecond group. The resulting row vector v ₂ having a length of n bits

${\overset{\rightarrow}{v}}_{2} = b_{1} \cdot {\overset{\rightarrow}{h}}_{1} + \ldots + b_{r} \cdot {\overset{\rightarrow}{h}}_{r}$

corresponds to the second row of the matrix M and (b₁,...,b_(n)) definesthe second row of the matrix J.

5. A linear combination v ₃ where v ₃ ≠ v ₁, v ₃ ≠ v ₂ and v ₃ ≠ (v ₁ +v ₂) is selected if possible from the first group, otherwise from thesecond group, and if even that is not possible, from the third group.The resulting row vector

${\overset{\rightarrow}{v}}_{3} = c_{1} \cdot {\overset{\rightarrow}{h}}_{1} + \ldots + c_{r} \cdot {\overset{\rightarrow}{h}}_{r}$

determines the third row of the matrix M and (c₁,...,c_(r)) defines thethird row of the matrix J.

6. The method is continued accordingly until all r rows of the checkmatrix M and the associated r rows of the matrix J have been determined.

This approach yields a hardware-optimized check matrix M. In this case,the matrix M is not uniquely determined; there are a large number ofsuch optimized matrices. When selecting a linear combination from agroup, there are a plurality of possibilities that lead to differentoptimized solutions. Each of the optimized check matrices H has the twoproperties (A)and (B).

The different possibilities for choosing the linear combinations from arespective group ultimately lead to different results for the generatedmatrices Mand J. This results in different inverse matrices J⁻¹. Inorder to find an efficient form for the matrix J⁻¹, a plurality of (orall) possibilities for choosing the linear combinations in therespective groups can be tried one after another. Each individual choicemade yields a pair of matrices (J,M). By way of example, that inversematrix J⁻¹ having the fewest ones per row can be used. The property (C)is fulfilled in this way.

Application Example The Error Correction Algorithm

The following matrix H indicated by way of example is the check matrixof a 2-error-correcting linear code having the length n= 15, thedimension k=7 and having the minimum distance d= 5.

$H = \begin{pmatrix}1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$

The columns of the matrix H are defined by the Boolean function

f(w, x, y, z) = w(x + y)

For j = 1,...,15 the j-th column of the matrix H is created inaccordance with the following specification:

-   1. Calculate the binary representation (j₀, j₁, j₂, j₃) of the    integer J.-   2. Calculate

k₀ = f(j₀, j₁, j₂, j₃),

k₁ = f(j₃, j₀, j₁, j₂),

k₂ = f(j₂, j₃, j₀, j₁),

k₃ = f(j₁, j₂, j₃, j₀).

The j-th matrix column is then given by

${\overset{\rightarrow}{h}}_{j} = \begin{pmatrix}j_{0} \\j_{1} \\j_{2} \\j_{3} \\k_{0} \\k_{1} \\k_{2} \\k_{3}\end{pmatrix}.$

The representability of the columns of the check matrix H by a uniformformula results in an efficient error correction algorithm with fast1-bit-error correction (comparable with the duration for a singlesyndrome calculation) and an accelerated 2-bit-error correction(computation duration corresponds approximately to that of 15 syndromecalculations). The error correction algorithm acquires as input thefollowing syndrome having a length of eight bits and defined by thecheck matrix H,

$S\left( \overset{\rightarrow}{y} \right) = H \cdot {\overset{\rightarrow}{y}}^{T},$

where y = (y₁,...,y₁₅) is the received data word.

Efficient Syndrome Calculation in Software

The matrix H is transformed into the canonical matrix K. For thispurpose, the column vectors at the positions 3,4,5,6,7,8 and 9 areresorted according to the permutation

σ = (3, 4, 8, 7, 5)(6, 9).

Afterward, a few elementary row operations are performed, such that the8×8 unit matrix arises at the beginning of the new matrix. In this case,an invertible 8×8 matrix L is obtained as a biproduct. The canonicalmatrix K results as:

$K = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$

There is the following relationship between the matrices H and K:

H = L ⋅ K ⋅ P,

where P is a 15 × 15 permutation matrix that represents the abovepermutation σ in matrix form. The matrices L and P are given by:

$L = \begin{pmatrix}1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$

and

$P = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}.$

The canonical matrix K is used for the syndrome calculation. For thispurpose, firstly the received data word y is resorted (“redistributed”)in accordance with the permutation σ or permutation matrix P indicatedabove:

$P \cdot \overset{\rightarrow}{y} = {{\overset{\rightarrow}{y}}^{\prime}}^{T}$

where

$\overset{\rightarrow}{y^{\prime}} = \left( {y_{1},y_{2},y_{4},y_{8},y_{3},y_{9},y_{5},y_{7},y_{6},y_{10},y_{11},y_{12},y_{13},y_{14},y_{15}} \right)^{T}.$

On the basis of the canonical matrix K, the syndrome

$Z = K \cdot {{\overset{\rightarrow}{y}}^{\prime}}^{T}.$

is determined. The syndrome calculation with the aid of the matrix K isable to be carried outin software more efficiently than with the use ofthe matrix H because the matrix K has the unit matrix I₈ (8×8 unitmatrix) at the beginning.

If the syndrome Z calculated using the matrix K for the vector y′ isequal to zero, the syndrome

$S\left( \overline{y} \right) = H \cdot {\overset{\rightarrow}{y}}^{T}$

is equal to zero and the received data word y is (with high probability)error-free.

Otherwise, the linear transformation L is applied to the syndrome Z,resulting in the syndrome

$S\left( \overset{\rightarrow}{y} \right) = H \cdot {\overset{\rightarrow}{y}}^{T} = L \cdot Z$

for the error vector calculation to be carried out.

Efficient Syndrome Calculation in Hardware

A check matrix M that is efficient in the case of a hardwareimplementation of the code for the syndrome calculation is determined asfollows, for example:

$M = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$

The matrix M contains 4 ones in each row, whereas the matrix H has rowshaving eight ones. The logical depth in the case of the calculation ofthe syndrome components with the matrix M is two.

By way of example, the first syndrome component w₁ of the M-syndromeresults in accordance with

w₁ = (y₁ ⊕ y₇) ⊕ (y₉ ⊕ y₁₅),

whereas the first syndrome component s₁ of the H-syndrome is determinedin accordance with

s₁ = [(y₁ ⊕ y₃) ⊕ (y₅ ⊕ y₇)] ⊕ [(y₉ ⊕ y₁₁) ⊕ (y₁₃ ⊕ y₁₅)]

On account of the smaller logical depth, the syndrome calculation withthe aid of the matrix M can be clocked more highly than the syndromecalculation with the aid of the matrix H.

The matrix M contains a total of 32 ones, whereas the matrix H has 48ones. The implementation costs (i.e., the number of gates) for thematrix M thus turn out to be ⅔ of the implementation costs for thematrix H.

If the syndrome is calculated with the matrix M, it has to betransformed by way of the linear transformation F into the syndrome H ·y ^(T) required by the error correction algorithm:

$H \cdot {\overset{\rightarrow}{y}}^{T} = F \cdot M \cdot {\overset{\rightarrow}{y}}^{T},$

where the linear transformation is determined by the matrix

$F = \begin{pmatrix}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}\mspace{6mu}\,.$

The matrices M and F were determined using the algorithm indicatedabove; the columns of the matrix H were not permuted in the process.Therefore, the matrices H and M define the same code vector space (whichis simultaneously the null space of the matrix H and the null space ofthe matrix M) and no permutation (redistribution) is required.

What is claimed is:
 1. A method for error correction, the methodcomprising: carrying out a syndrome calculation on the basis of a dataword, with a matrix M on the basis of a matrix H of a code, transformingthe result of the syndrome calculation by means of a linear mapping, inresponse to the result of the syndrome calculation revealing that thedata word is erroneous, determining an error vector on the basis of theresult of the linear mapping by means of an efficient error correctionalgorithm, and correcting the erroneous data word on the basis of theerror vector.
 2. The method of claim 1, wherein the matrix M used forthe syndrome calculation has the following properties: each row of thematrix M has the fewest possible ones, the matrix M overall has thefewest possible ones.
 3. The method of claim 1, wherein the methodcomprises determining the matrix M used for the syndrome calculation onthe basis of the matrix H of the code by: determining linearcombinations for the row vectors of the matrix H, determining a Hammingweight for each linear combination, determining row vectors of thematrix Mon the basis of the linear combinations prioritized according totheir rising Hamming weight, coefficients of the linear combinationsrespectively selected determining rows of a matrix J.
 4. The method ofclaim 3, further comprising determining non-trivial linear combinationsfor the row vectors of the matrix H, determining the Hamming weight isdetermined for each linear combination, grouping the linear combinationsaccording to their Hamming weight, selecting linear combinations in turnwith rising Hamming weight, determining a row of the matrix M i on thebasis of each selected linear combination and determining a row of thematrix J on the basis of the coefficients of the selected linearcombination, and selecting the linear combinations with rising Hammingweight until all the rows of the matrix M and all the rows of the matrixJ have been determined.
 5. The method as claimed in claim 4, wherein thelinear mapping is based on the inverse matrix J⁻¹.
 6. The method asclaimed in claim 5,wherein the matrix J⁻¹ is determined such that it hasthe fewest possible ones per row.
 7. A device for error correctioncomprising a processing circuit, wherein the processing circuit isconfigured to: on the basis of a data word, carry out a syndromecalculation with a matrix M on the basis of a matrix H of a code,determining whether the result of the syndrome calculation reveals thatthe data word is erroneous, responsive to determining the result of thesyndrome calculation reveals that the data word is erroneous, transformthe result of the syndrome calculation by means of a linear mapping,determine, on the basis of the result of the linear mapping, an errorvector by means of an efficient error correction algorithm, and correctthe erroneous data word on the basis of the error vector.
 8. The deviceof claim 7, wherein the processing circuit is configured such that nocorrection is effected if the fact that no error was detected wasdetermined on the basis of the result of the syndrome calculation. 9.The device of claim 7, wherein the matrix M used for the syndromecalculation has the following properties: each row of the matrix M hasthe fewest possible ones, the matrix M overall has the fewest possibleones.
 10. The device of claim 7, wherein the processing circuit isconfigured to determine the matrix M used for the syndrome calculationon the basis of the matrix H of the code by: determining linearcombinations for the row vectors of the matrix H, determining a Hammingweight for each linear combination, determining row vectors of thematrix Mon the basis of the linear combinations prioritized according totheir rising Hamming weight, coefficients of the linear combinationsrespectively selected determining rows of a matrix J.
 11. The device ofclaim 10, wherein the processing circuit is further configured to:determine non-trivial linear combinations for the row vectors of thematrix H, determine the Hamming weight for each linear combination,group the linear combinations according to their Hamming weight, selectlinear combinations in turn with rising Hamming weight, a row of thematrix Mbeing determined on the basis of each selected linearcombination and a row of the matrix J being determined on the basis ofthe coefficients of the selected linear combination, select the linearcombinations with rising Hamming weight until all the rows of the matrixM and all the rows of the matrix Jhave been determined.
 12. The deviceas claimed in claim 11, wherein the processing circuit is configuredsuch that the linear mapping is based on the inverse matrix J⁻¹.
 13. Thedevice as claimed in claim 12, wherein the processing circuit isconfigured such that the matrix J⁻¹ is determined such that it has thefewest possible ones per row.
 14. A non-transitory computer-readablemedium comprising, stored thereupon, a computer program product that isdirectly loadable into a memory of a digital computer and that comprisesprogram code parts configured to cause the digital computer to carry outsteps of a method for error correction comprising: carrying out asyndrome calculation on the basis of a data word, with a matrix M on thebasis of a matrix H of a code, transforming the result of the syndromecalculation by means of a linear mapping, in response to the result ofthe syndrome calculation revealing that the data word is erroneous,determining an error vector on the basis of the result of the linearmapping by means of an efficient error correction algorithm, andcorrecting the erroneous data word on the basis of the error vector.